Theory and Applications of Fractional Order SystemsView this Special Issue
Finite Series Representation of the Inverse Mittag-Leffler Function
The inverse Mittag-Leffler function is valuable in determining the value of the argument of a Mittag-Leffler function given the value of the function and it is not an easy problem. A finite series representation of the inverse Mittag-Leffler function has been found for a range of the parameters and ; specifically, for and for . This finite series representation of the inverse Mittag-Leffler function greatly expedites its evaluation and has been illustrated with a number of examples. This represents a significant advancement in the understanding of Mittag-Leffler functions.
The Mittag-Leffler function is defined by the power series  While the argument and the parameters and can in general be complex provided Re , in this work , , and will be restricted to those values most commonly found in physical problems; namely, the argument will be restricted to real numbers and and will be restricted to positive real numbers. The Mittag-Leffler function is a generalization of the exponential function and arises frequently in the solutions of differential and/or integral equations of fractional (noninteger) order in much the same way as the exponential function appears in solutions of differential equations of integer order. Thus, Mittag-Leffler functions play a fundamental role in the theory of fractional differential equations. Consequently, books devoted to the subject of fractional differential equations (i.e., Podlubny , Kilbas et al. , and Diethelm ) all contain sections on the Mittag-Leffler functions. In addition to their inherent mathematical interest, Mittag-Leffler functions are also important in theoretical and applied physics and all the sciences (i.e., Hilfer , Mainardi , and Magin ). The works of Mainardi and Gorenflo , Magin , Berberan-Santos , Gupta and Debnath , and Haubold et al.  are a few of the numerous articles also worth noting.
The inverse Mittag-Leffler function is defined as the solution of (2)  Despite the inherent importance of Mittag-Leffler functions in fractional differential equations, with the wealth of analytical information about , the inverse has been largely unexplored. The one exception is the excellent work of Hilfer and Seybold  who have determined its principal branch numerically.
The power series representation of any Mittag-Leffler function can be inverted yielding an infinite series for the inverse. However, these infinite series are slow to converge and terminating the series always introduces error which is hard to evaluate. This present work identifies regions in the domain of and where the inverse of the Mittag-Leffler function can be written as a finite series. This represents the first time the inverse Mittag-Leffler function has been written as a finite series as opposed to an infinite series which greatly expedites its evaluation. Before deriving these expressions for the inverse Mittag-Leffler function, a brief review of the theory of power series and their inverses is in order.
Consider the convergent series which expresses the function in terms of powers of with the corresponding coefficients given by The inversion of the function requires only the sole assumption that . That is, there exists one and only one function which represents the inverse of the , which is expressible by a convergent power series of the form  The process of finding the series expansion for is called reversion of the series. The coefficients can be determined in terms of the coefficients by substituting (3) into (4) and equating coefficients of like powers of on both sides of the equation yielding The coefficients can be found in the literature [15–17]. An explicit expression for the coefficients can be derived using the Lagrange inversion theorem. If is analytic at and , then the inverse of exists and is analytic about . Furthermore, if , the Lagrange inversion theorem gives the Taylor series expansion of the inverse function as  The coefficients are determined by comparing (6) and (4) yielding Substituting from (3) yields Factoring out in (8) and defining yields Using the multinomial expansion and performing the required differentiation yields the desired result  where and the numbers are nonnegative integers and the summation extends over all partitions of . For example, contains 5 terms since the Diophantine equation has 5 integer solutions or partitions. The number of partitions for is 42; for there are 204226 partitions and for the number of partitions is 190569292. Consequently, the explicit tabulation of the full expression for the coefficients rapidly becomes a rather tedious task. Nevertheless, the coefficients are given in Table 1. An equivalent expression for the general term in the reversion of series is given in a different form by McMahon .
By an appropriate change of variables it is always possible to write the power series in a form which results in simplified expressions for the coefficients in the reversed power series. Equation (3) can be rewritten as Defining the new variables , , , and so forth, (11) becomes and the reversed series is given by The resulting coefficients for , and 4 are given by These can be shown to be equivalent to (5) by setting and . The coefficients for can be found tabulated in , for in  and for they are tabulated in  with a different choice of the sign of . Müller  has reported an alternative expression for and some symmetry relations for the coefficients.
3. Application to Mittag-Leffler Functions
Many Mittag-Leffler functions can be represented in terms of elementary functions. For example, Applying (10) from the above theory to these functions whose values can be determined as accurately as possible using their alternative representations yields A few observations are in order. Equations (16) and (17) are typical of the inverse of most infinite series; that is, they are also infinite series and do not converge rapidly. This can be easily illustrated by the following examples. For , (16) should yield (equivalently ). However, (16) requires 20 terms before the value of is as large as 0.99999 (5 nines), 44 terms for 10 nines, 68 terms for 15 nines, and 92 terms for 20 nines. Whereas for , where (16) should yield , 156995 terms are required before the value of is as large as 9.9999 (5 nines), 391895 terms for 10 nines, 635259 for 15 nines, and 881815 terms for 20 nines. Similarly, for where (16) should yield , 16730862 terms are required before the value of is as large as 14.999 (3 nines), 51041531 terms for 8 nines, 87009540 terms for 13 nines, and 123532970 terms for 18 nines. For , as becomes large (or equivalently ), the number of terms in (16) required to yield a value accurate to a given number of significant digits becomes astronomically large.
A similar behavior is exhibited in (17). For , (17) should yield . To obtain a value of as large as 0.99999 (5 nines), 12 terms are required, 24 terms for 10 nines, 36 terms for 15 nines, and 48 terms for 20 nines. For , (17) should yield , but 81 terms are required to obtain a value of as large as 9.9999 (5 nines), 162 for 10 nines, 243 terms for 15 nines, and 324 terms for 20 nines. For , (17) should yield , but 770 terms are required to obtain a value of as large as 99.999 (5 nines) and 1540 terms for 10 nines. There is, however, one big difference between (16) and (17). Equation (16) is one of the few inverses of a Mittag-Leffler function, where the coefficients in the inverse given in (10) and itemized in Table 1 for simplify to a tractable expression; in this case . The mathematical manipulations required to obtain the coefficients in (17) using (10) become algebraically intensive as becomes large. Whereas given in Table 1 contains 101 terms, contains more than terms. Consequently, although the infinite series given in (17) correctly represents the inverse Mittag-Leffler function, it is impractical to use for anything other than small where only a reasonable number of terms are needed for the required accuracy. This is the case for most of the inverse Mittag-Leffler functions.
It is obvious in looking at the coefficients in Table 2 that they are approaching a constant as becomes large. In this case, the constant is . Subsequently, the first 20 significant digits for all coefficients after are identical differing only after the first 20 digits. Thus, applying (4) with , , the inverse for can be written as Equation (18) assumes that all coefficients for can be approximated by . The approximation is valid provided that an answer accurate to no more than 20 significant digits is sufficient. The last term in (18) is a geometric series which can be replaced by its corresponding sum yielding Equation (19) represents a finite series for the inverse Mittag-Leffler function for or equivalently accurate to 20 significant digits. The series has been tested numerically and in all cases tested gives the correct answer to at least 20 significant digits or equivalently . This finite series representation of the inverse Mittag-Leffler function has at least 3 advantages over the infinite series representation: (1) the finite series greatly expedites the evaluation of the inverse, (2) it is not limited to small , and (3) there is no ambiguity concerning the number of terms needed in the series to obtain a required accuracy in the final answer.
Note that if the required accuracy is only 10 significant digits, the first 10 digits of the coefficients after are identical differing only after the first 10 digits. In this case, the equation for the inverse can be written as The fact that the coefficients approached a constant as becomes large allowed the infinite series to be written as a finite series. For what other Mittag-Leffler functions do the coefficients in the inverse approach a constant?
4. Inverse Mittag-Leffler Functions for Which Approach a Constant
Evaluation of great many inverse Mittag-Leffler functions reveals several important points. (1) It has been shown that the Mittag-Leffer function with these and parameters, namely, and , is a completely monotonic decreasing function [24, 25], and thus the inverse is guaranteed to be single valued. (2) The coefficients in the inverse approach a constant only when the parameter is either 1 or 2. (3) The coefficients approach a constant only when the parameter . (4) The coefficients approach a constant given by Consequently, as , the coefficient for both and 2. However, for the coefficient is always less than 1 while for , is always greater than 1 as . (5) The smaller the value of , the fewer the numerical terms required in the inverse series to obtain a given significant digit accuracy. This is illustrated in Table 3 which gives the number of terms required in the finite representation of the inverse Mittag-Leffler function for 20-significant-digit accuracy for various values of with and .
Extending this logic to its natural conclusion implies that at no terms will be required in the series. To see that this is correct, note that using (1) both and reduce to when . Inverting and solving for yield . This is consistent with (19) which reduces to this same result when the upper limit on the summation is (no terms in the summation) and the factor is replaced by the more general equation (21) which gives unity for and or .
Conversely, as approaches 1, an increasingly larger number of numerical terms are required in the inverse series to obtain a given significant digit accuracy as Table 3 illustrates. (6) Consequently, as increases above 1/2, the inverse Mittag-Leffler function described by a finite series requires more and more terms becoming less practical. For example, for and , for to converge to just 5 significant digits requires 2215 terms while, for and , requiring 1828 terms for the same convergence. (7) For the same , the number of terms in the inverse for a desired accuracy is less for than for . (8) According to (21), when and , the coefficients in the inverse for the Mittag-Leffler function approach the constant zero as as seen in (16) while for and the coefficients in the inverse for the Mittag-Leffler function approach 1 as . (9) As noted above, according to (21), for the coefficients as approach 1 as and as and is greater than 1 for . This implies that there exists a relative maximum value of as in the range . This maximum occurs at and corresponds to as . Illustrating the above observations are numerous examples in the next section.
5. Results for Specific and
In this section, specific examples of various inverse Mittag-Leffler functions calculated using (10) will be given. Since the number of terms in the finite series for the inverse increases dramatically for , then all examples will be for . All equations for the inverses are written assuming a desired 20-significant-digit accuracy. This is far greater accuracy than most requirements might call for; however, the equations can then be easily modified to any degree of accuracy less than 20 as outlined in the discussion of (20). Each Mittag-Leffler inverse example includes the equation of the form given in (19) valid for (equivalently ) representing the finite series representation of the inverse and a table with the corresponding coefficients truncated to 20 significant digits. The specific values of and in each example are itemized in Table 4 which includes references to the corresponding equations and table numbers for each example inverse.
For and , the equation for the inverse is given by where and the coefficients are given in Table 5.
For and , the equation for the inverse is given by where and the coefficients are given in Table 6.
For and , the equation for the inverse is given by where and the coefficients are given in Table 7.
For and , the equation for the inverse is given by where and the coefficients are given in Table 8.
For and , the equation for the inverse is given by where and the coefficients are given in Table 9.
For and , the equation for the inverse is given by where and the coefficients are given in Table 10.
For and , the equation for the inverse is given by where and the coefficients are given in Table 11.
For and , the equation for the inverse is given by where and the coefficients are given in Table 12.
For and , the equation for the inverse is given by where and the coefficients are given in Table 13.
For and , the equation for the inverse is given by where and the coefficients are given in Table 14.
For and , the equation for the inverse is given by where and the coefficients are given in Table 15.
For and , the equation for the inverse is given by where and the coefficients are given in Table 16.
For and , the equation for the inverse is given by where and the coefficients are given in Table 17.
For and , the equation for the inverse is given by where and the coefficients are given in Table 18.
For and , the equation for the inverse is given by where and the coefficients are given in Table 19.
A finite series representation of the inverse Mittag-Leffler function has been found for a range of the parameters and ; specifically for and for . Various properties of the coefficients in the finite series have been examined. In addition, a formula for as is established and the limiting cases were investigated. These properties are illustrated in 16 examples of inverse Mittag-Leffler functions. Determining the value of the argument of a Mittag-Leffler function given the value of the function is not an easy problem and the finite series representation of the inverse Mittag-Leffler function greatly expedites their evaluation and represents a significant advancement.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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